The problem given is a sequence defined by:

$a_n = \frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \dots + \frac{n}{2^n}$

This sequence can be written as a sum:

$a_n = \sum_{k=1}^{n} \frac{k}{2^k}$

To analyze and possibly find a closed-form expression for $a_n$, let's break down the components.

Step-by-Step Solution Outline

1. Term Identification: Each term in the series has the form $\frac{k}{2^k}$, where $k$ is the term index.
2. Approach: Such series can often be evaluated by finding a generating function or using calculus techniques if possible, but in some cases, they do not simplify nicely.
3. Known Result: A common result for sums involving terms like $\frac{k}{r^k}$ (with $r > 1$) can sometimes be derived using known formulas for geometric or arithmetic-geometric series.

Would you like the detailed steps for finding the closed form or just an approximation of the series sum for large $n$?